 Most partial differential equations originate from one of about three different problems. The heat equation is one of those problems. Suppose we have a metal bar of length L, where we assume that the endpoints are heated. The bar is perfectly insulated along its length. Under these conditions, the temperature u at any point x along the bar at time t will satisfy the one-dimensional heat equation, where alpha is a physical constant assumed to be real. To solve the heat equation, we assume a solution of the form u of t of x is the product of a function of t only with a function of x only. Remember, we're allowed to do this because it is easier to obtain forgiveness than permission. If this actually leads to a solution to the heat equation, the existence of that solution justifies this assumption. So let's find our partial derivatives. We have the second partial derivative of u with respect to x, and the partial of u with respect to t. So our heat equation becomes, and we'll do a little algebra to get it into this form, where we separated the t functions from the x functions. Now here's the important thing to notice. The right-hand side depends only on x, and the left-hand side depends only on t. So why is this significant? In order for these to be equal for all values of t and x, this means that both sides must be constant. Otherwise, we could hold one variable constant and change the other, changing the value of one side of the equation. So we'll designate the constant lambda and call it the separation constant. What's that? Oh, wait. Sorry. A portal in the spacetime continuum opened up, and by future self-informed me, it's actually better if we call this constant minus lambda. So we'll make all of these equal to minus lambda. So let's solve these differential equations. The first one gives us a solution, and we can rewrite this in operator form, and we have our solutions. Similarly, our second differential equation can be rewritten, rewriting in operator form, which gives us roots of plus or minus square root of minus lambda. So our solutions will be, and we end up with our function looking like this, where we'll adjust to writing this in exponential or trigonometric form depending on whether square root minus lambda is real or complex. And so this gives us a solution to the heat equation, where our g function is an exponential, and our f function is an exponential or trigonometric function. It's worth taking a moment to look at our equation and remember consistency counts. Newton's law of cooling led to a temperature function of the form C e to power minus kt, which is an exponential function, and the time-dependent part of our solution looks like Newton's law of cooling, which suggests we're on the right track. So far so good. But notice our solutions depend on lambda, and we don't know lambda. Well, actually, that's not really a problem. In fact, any value of lambda gives us a solution. Since any value of lambda gives us a solution, this leads to the following. Suppose we pick some values for lambda, giving a solution to u1, u2, and so on. Any linear combination of u1 and u2 will also solve the heat equation. You should probably prove that statement. This leads to the following. The one-dimensional heat equation has a series solution where g functions are exponentials, and our f functions are either exponential or trigonometric, depending on the values of lambda, for every lambda in some set, capital lambda. While our theorem means that we don't have to worry about the value of lambda, it leads to a very different problem. Since every value of lambda will give a solution, how can we find the solution to a specific boundary value problem? We'll take a look at that next.